With the IRS scandal over targeting conservative groups still simmering, two senators — one from each party — say the time is ripe to push the beleaguered agency to go after a real, proven target: federal employees who are tax cheats.

Sens. Tom Coburn, Oklahoma Republican, and Mark L. Pryor, Arkansas Democrat, have introduced a bill that would allow federal agencies to fire any employee with “seriously delinquent” tax debt — outstanding debt for which a notice of lien has been filed in public records. Anyone with unpaid federal taxes — which could include hundreds of thousands of government and Postal Service workers on the federal payroll — would be ineligible for employment with the federal government.

Exceptions would be made for tax debt being repaid pursuant to an agreement with the Internal Revenue Service and tax debt subject to a collection due-process hearing.

“Instead of using the power of the state to silence and intimidate political opponents, the IRS should be focused on collecting taxes from people who are delinquent,” said Mr. Coburn, referring to recent revelations of the improper IRS targeting of tea party and other conservative groups seeking tax-exempt status from 2010 to 2012.

The Republican lawmaker said that while most federal employees are “hard-working and responsible citizens,” it’s wrong to force law-abiding taxpayers to fund the paychecks of a minority of federal workers who “don’t feel like paying their taxes.”

Mr. Pryor added that the “common-sense bill” is needed in today’s wobbly economy to “ensure that we’re getting the most bang for our buck.”